Custom Component ignoring Template Style

I'm working on a custom component for a site that also uses a custom
template I also built. The template uses parameters to modify the overall
layout (including a setting that disables two optional columns), and works
beautifully with the content system and other core Joomla systems.
However, when I tell a menu item which renders a page from the custom
component that is should use a stored alternate template style, that page
always goes back to rendering the default style settings.

The menu item contains a template_style_id value referring to the correct
DB record, and the DB record for that style contains the proper values. If
I change the default style's settings, the custom component renders the
correct layout. But if I select an alternate style, it goes back to using
the default settings.

I don't know if the issue is within my component or my template, and I have
no clue where to look. Any suggestions where to start troubleshooting this?

Thanks.